<header>
    证明
</header>
<p>
    <span class="title">
        定理
    </span>
    设α<sub>1</sub>,α<sub>2</sub>,...,α<sub>r</sub>与β<sub>1</sub>,β<sub>2</sub>,...,β<sub>s</sub>是两个向量组，如果
</p>
<ol>
    <li>
        向量组α<sub>1</sub>,α<sub>2</sub>,...,α<sub>r</sub>可以经β<sub>1</sub>,β<sub>2</sub>,...,β<sub>s</sub>线性表出；
        <span class="toRight">①</span>
    </li>
    <li>
        r＞s。
        <span class="toRight">②</span>
    </li>
</ol>
<p>
    那么向量组α<sub>1</sub>,α<sub>2</sub>,...,α<sub>r</sub>必线性相关。
</p>
<h2>
    证：
</h2>
<p>
    由①式可知，必有
    <span class="oneline">
        α<sub>j</sub>=
        <code>
            ["sum","i=1","s",""]
        </code>
        β<sub>i</sub>
        k<sub>ij</sub>
        ，i=1,2,...,s
    </span>
    而由线性相关的定义可知，α<sub>1</sub>,α<sub>2</sub>,...,α<sub>r</sub>线性相关的等价描述为：存在x<sub>1</sub>、x<sub>2</sub>、...、x<sub>r</sub>使得：
    <span class="oneline">
        <code>
            ["sum","j=1","r",""]
        </code>
        x<sub>j</sub>
        α<sub>j</sub>
        =
        <code>
            ["sum","j=1","r",""]
        </code>
        x<sub>j</sub>
        <code>
            ["sum","i=1","s",""]
        </code>
        β<sub>i</sub>
        k<sub>ij</sub>
        =
        <code>
            ["sum","i=1","s",""]
        </code>
        <code>
            ["sum","j=1","r",""]
        </code>
        x<sub>j</sub>
        k<sub>ij</sub>
        β<sub>i</sub>
        =0
        <span class="toRight">③</span>
    </span>
</p>
<p>
    如果对所有的i=1,2,...,s都有
    <span class="oneline">
        <code>
            ["sum","j=1","r",""]
        </code>
        x<sub>j</sub>
        k<sub>ij</sub>=0
        <span class="toRight">④</span>
    </span>
    那③式也就得证了。现在，我们用把④写开成方程组的形式：
</p>
<p>
    <code>
        ["equationSet",
            ["join",["rightBottom","k","11"],["rightBottom","x","1"]," + ",["rightBottom","k","12"],["rightBottom","x","2"]," + ... + ",["rightBottom","k","1r"],["rightBottom","x","r"]," = 0"],
            ["join",["rightBottom","k","21"],["rightBottom","x","1"]," + ",["rightBottom","k","22"],["rightBottom","x","2"]," + ... + ",["rightBottom","k","2r"],["rightBottom","x","r"]," = 0"],
            "                                               ... ... ... ...",
            ["join",["rightBottom","k","s1"],["rightBottom","x","1"]," + ",["rightBottom","k","s2"],["rightBottom","x","2"]," + ... + ",["rightBottom","k","sr"],["rightBottom","x","r"]," = 0"]
        ]
    </code>
</p>
<p>
    s个约束条件，r个未知量，而r＜s，故方程组必有解，即此定理得证。
</p>